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In Gapeev and K\"uhn (2005), the stochastic game corresponding to perpetualconvertible bonds was considered when driven by a Brownian motion and acompound Poisson process with exponential jumps. We consider the samestochastic game but driven by a spectrally positive L\'evy process. Weestablish a complete solution to the game indicating four principle parameterregimes as well as characterizing the occurence of continuous and smooth fit.In Gapeev and K\"uhn (2005), the method of proof was mainly based on solving afree boundary value problem. In this paper, we instead use fluctuation theoryand an auxiliary optimal stopping problem to find a solution to the game.

The Gapeev-K\"uhn stochastic game driven by a spectrally positive L\'evy process